a + 3)(a^3 - 2a^2 + 5a -2)

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A L G E B R A

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14

MULTIPLYING OUT

THE DISTRIBUTIVE RULE

m(a + b) = ma + mb

"To multiply a sum, multiply each term of the sum."

That is called the distributive rule.  m multiplies a, then it multiplies b.  We say that m is "distributed" to a and b.

Example 1.    2(x + y + z) = 2x + 2y + 2z.

We have distributed 2 to x, y, and z.  We have "multiplied out."

Example 2.    3x4(x² − 5x + 1) = 3x6 − 15x5 + 3x4

That is,

3x4· x² = 3x6,   Rule 1 of exponents (Lesson 13)
 
3x4· −5x = −15x5
 
3x4· 1 = 3x4

Problem 1.    −1(ab + cd)

What will be the effect of multiplying by −1?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

Every sign will change.  −1(ab + cd) = −a + bc + d

Therefore in an equation, we may change all the signs on both sides. This equation

  x + ab  =  c
 
implies this one:
 
  xa + b  =  c.

Theoretically, we have multiplied both sides by −1.

Problem 2.   Multiply out.

   a)   5(x + 4) = 5x + 20      b)   5(x − 4) = 5x − 20
 
  c)   x(x + 1) = x² + x     d)   2x(3x² + 5x − 6) = 6x3 + 10x² − 12x
 
  e)   −5x4(x3 − 4x² + 2x − 6) = −5x7 + 20x6 − 10x5 + 30x4  

f)   2xy(x² − 3xy + y²)  = 2x3y − 6x²y² + 2xy3

g)   −4xy²(x3y − 6xy² − 2x + 3y + 1)

= −4x4y3 + 24x²y4 + 8x²y² − 12xy3 − 4xy²

Problem 3.   Multiply out and simplify, that is, add the like terms.

  a)   2(4x + 5y) + 3(5xy)   =   8x + 10y + 15x − 3y
 
    =   23x + 7y
  b)   4(2x − 1) − 5(x − 2)   =   8x − 4 − 5x + 10
 
    =   3x + 6
  c)   3x(3x − 2y) − 2y(xy)   =   9x² − 6xy − 2yx + 2y²
 
    =   9x² − 8xy + 2y²
  d)   x(x² − 10x + 25) − 5(x² − 10x + 25)
 
  =  x3 − 10x² + 25x − 5x² + 50x − 125
 
  =  x3 − 15x² + 75x − 125
  e)   a(a² − 2ab + b²) − b(a² − 2ab + b²)
 
  =  a3 − 2a²b + ab² − ba² + 2ab² − b3
 
  =  a3 − 3a²b + 3ab² − b3

A sum by a sum

(a + b + c)(x + y + z)

First distribute a  to x, y, and z.

Then distribute b.

Then distribute c.

(a + b + c)(x + y + z)

= ax + ay + az + bx + by + bz + cx + cy + cz

Problem 4.   Multiply  (pq)(xy + z).   Observe the Rule of Signs (Lesson 4).

(pq)(xy + z) = pxpy + pz  −  qx + qyqz

Example 3.   Multiply out  (x2)(x + 3).  Simplify by adding the like terms.

Solution.   First distribute x, then distribute −2:

(x2)(x + 3) = x· x + x· 3  − 2· x2· 3
 
  = x² + 3x2x − 6
 
  = x² + x − 6

The student should not have to write the first line, but should be able to write the second line --

x² + 3x2x − 6

-- immediately.

Problem 5.   Multiply out.  Always simplify by adding the like terms.

  a)   (x + 5)(x + 2)   =   x² + 2x + 5x + 10
 
    =   x² + 7x + 10
  b)   (x + 5)(x − 2)   =   x² − 2x + 5x − 10
 
    =   x² + 3x − 10
  c)   (x − 5)(x − 2)   =   x² − 2x − 5x + 10
 
    =   x² − 7x + 10
  d)   (2x − 1)(x + 4)   =   2x² + 8xx − 4
 
    =   2x² + 7x − 4
  e)   (3x + 2)(4x − 5)   =   12x² − 15x + 8x − 10
 
    =   12x² − 7x − 10
  f)   (5x − 1)²   =  (5x − 1) (5x − 1)
 
   =  25x² − 5x − 5x + 1
 
   =  25x² − 10x + 1
  g)   (6x + 1)(6x − 1)  =  36x² − 6x + 6x − 1
 
   =  36x² − 1
Example 4.   (x − 4)(x² + 3x − 10) = x3 + 3x²  − 10x
 
    − 4x²  − 12x + 40
 
  = x3x²  − 22x + 40

Notice:  Upon distributing −4, we have anticipated the like terms by aligning them.  However, that is not strictly necessary.

Problem 6.   Multiply out.

  a)  (x + 2)(x² + 4x − 5) = x3 + 4x²  − 5x
 
    + 2x²  + 8x − 10
 
  = x3 + 6x²  + 3x − 10
  b)  (x − 3)(x² − 6x + 9) = x3 − 6x²  + 9x
 
    − 3x²  + 18x − 27
 
  = x3 − 9x²  + 27x − 27
  c)  (3x − 4)(x² − 7x − 2) = 3x3 − 21x²  − 6x
 
    − 4x²  + 28x + 8
 
  = 3x3 − 25x²  + 22x + 8
  d)  (x − 1)(x3 + x² + x + 1) = x4 + x3  + x² + x
 
    x3  x² − x − 1
 
  = x4 − 1

Note:  The effect of multiplying by x is to increase each exponent by 1; the effect of multiplying by −1 is to change each sign.


Next Lesson:  Common Factor


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